TUYMAADA 2015

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$ . Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ A. Golovanov

There is some natural number $n>1$ on the board. Operation is adding to number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by $3^{2000}$

A. Golovanov